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        general >> complex analysis >> point of symmetry
        
(Message started by: trusure on Apr 20^th, 2009, 5:19pm)

Title: point of symmetry
Post by trusure on Apr 20^th, 2009, 5:19pm

while I was trying to find a Linear Fractional Transformation from the upper half plane to itself, with 0 mapped to 1 and i mapped to 2i, I used that the point of symmetric of i is -i mapped to -2i which is the point of symmetric for 2i,(using the symmetric principle), so now I have 3 pints mapped to 3 points, and using the cross ratio I found f(z)=2.5 *z +1 ! which is not correct ??
so, where is my mistake ??

can anyone help me ?

Thank you

Title: Re: point of symmetry
Post by Eigenray on Apr 20^th, 2009, 8:11pm

Probably not, if you don't show what you did. Solving

(z, 0; i, -i) = (f(z), 1; 2i, -2i)

for f(z) should work.

Title: Re: point of symmetry
Post by trusure on Apr 20^th, 2009, 8:56pm

does it matter the order of the points while using the cross ratio ?!!

Title: Re: point of symmetry
Post by Eigenray on Apr 20^th, 2009, 9:26pm

No, as long as you put the points in the same order as their images. E.g.,

(-i, 0; z, i) = (-2i, 1; f(z); 2i)

would also work, but not

(z, 0; i, -i) = (f(z), 2i; -2i, 1)